Abstract: It is shown that every Lie algebra can be represented
as a bivector algebra; hence every Lie group can be represented
as a spin group. Thus, the computational power of geometric algebra
is available to simplify the analysis and applications of Lie groups
and Lie algebras. The spin version of the general linear group is
thoroughly analyzed, and an invariant method for constructing real
spin representations of other classical groups is developed. Moreover,
it is demonstrated that every linear transformation can be represented
as a monomial of vectors in geometric algebra.